During our investigations on fractional calculus [1]} we met the integral
\(I=\int _0^1\dfrac{(1-t)^\alpha (1+t)^\beta }{t^\lambda }P_n^{(\alpha ,\beta )}(t)dt \qquad \text{with}\qquad \lambda >0\ \ \text{and}\ \ \operatorname{Re}(\alpha ,\beta )>-1\) 
